Electro-optic response modelling

6.3 Electro-optic response modelling

6.3.1 Electric-field induced changes of optical properties

The description of the influence of the electric field on the changed optical properties is much more complex than introduced in Sec. 4.1. Instead of a homogeneously distributed electric field with only a single component, strongly inhomogeneous distributions are found for all field components. Therefore, the influence of all three electric-field components has to be considered for a given domain orientation.

As already described in Sec. 4.1, the altered optical properties are given by the changed elements of the inverse dielectric tensorη according to

∆ηij = ∆ε⁻¹_ij = r^S_ijk+p^E_ijmndkmn

Ek=r^eff_ijkEk. (6.7) If a crystal having 4mm symmetry such as BaTiO₃ is considered, all occurring changes can be expressed in the crystal coordinates as

∆η_ij =

with n_o denoting the ordinary and n_e the extraordinary refractive index at zero electric field.

In case of ac domain, the crystal coordinate system (123) coincides with the coordinates used in the model system (xyz) and the electric-field components can be directly identified asE₁=E_x,E₂ =E_y, andE₃=E_z. For anadomain, we find an additional transformation between the coordinates of the model and the crystal, because of the different orientation.

This transformation T_ij is simply given by

Tij =

transforming a vector from the model system to the crystal system and vice versa. This transformation maps the electric-field components in accordance withE1 =−E_z,E2 =Ey, and E3 =Ex.

Since optical fields must be described in the model coordinates for the numerical calcula-tions, the resulting η matrix has to be transformed to the model coordinates, too, and is then given by

It it obvious that additional transformations are needed to describe the electro-optic re-sponse of inverted domains in the same way. These transformations are reported in the appendix A.1 and leads to a very simple result: The electro-optic response of inverted domains can be described in terms of the response of the non-inverted ones by inverting the signs of all electric-field components. This result is valid for both aand cdomains.

The next step for modelling the electro-optic measurements is to transform the changed optical properties, as given by the η matrix, into a change of transmitted light power, which is the quantity recorded in the experiment. For this, Fresnel’s formula is applied to calculate the local transmittance for each addressed point on the sample surface, which we do by simply using the electro-optically changed refractive index ∆ns directly at the surface. In a more accurate treatment, one would have to calculate an effective change

∆nef f for the given depth profile of ∆n. However, this value comes close to ∆ns, provided that the decay of ∆n into the sample is not too steep. In the present case, this is fairly well fulfilled as illustrated by Fig. 6.5, which shows ∆n_{ef f}/∆n_s for an exponential ∆n profile, as calculated forλ= 633 nm and normal incidence [83]. From this, values of 96%

and 74%, respectively, rather close to 100% for an a and a c domain can be estimated, which demonstrates that our simplified treatment is satisfactory.

0 25 50 75 100 125 150

Fig. 6.5 Quality of the approximation by Fresnel’s formula for a spatially nonuniform change of the refractive index. The curve was calculated for an exponentially decaying change of the refractive index [83] at a wavelength λ= 633 nm and shows the effective value of the change.

The accuracy of the approximation for the electro-optic-response mod-elling of a and c domain areas is reflected by the effective changes of 96% and 74%.

6.3. Electro-optic response modelling 63

Table 6.1 Probed electro-optic coefficients and corresponding field components with respect to the selected orientation ~u of optical polarization.

~

u= (ux, uy, uz) probedηuv probed rijkEw

adomain cdomain

(1,0,0) ηxx r333Ex r113Ez

(cosφ,0,sinφ) ηxx r333Ex r113Ez

tilt in xz plane η_zz r₁₁₃E_x r₃₃₃E_z ηxz, ηzx −r₁₃₁Ez r131Ex

(0,1,0) ηyy r333Ex r113Ez

(0,cosγ,sinγ) ηyy r113Ex r113Ez

tilt in yz plane η_zz r₁₁₃E_x r₃₃₃E_z η_yz, η_zy 0 r₁₃₁E_y

As incidence of the light can be assumed to be approximately normal in our model, the transmission coefficientT(n) is simply given by

T(n) = 4n

(n+ 1)² . (6.12)

With reference to Sec. 4.1, the refractive index n can be calculated for normal modes of optical polarization from

n(~u) = (ηijuiuj)^−1/2, (6.13) where~uis the unit vector along the polarization direction of the optical field. By selecting a proper~u, different elements of the η matrix can be probed. This way, different electro-optic coefficients and electric-field components can be addressed as listed in Tab. 6.1.

To calculate the total transmitted power, assumptions about the incident light have to be made. The light radiated from an etched fiber tip is characterized by a Gaussian-like focus close to the tip apex, as reported in Sec. 3.2.2. For modelling the incident light, the intensity I is therefore approximated by a Gaussian distribution given by

I =I0exp

−2r² w²_b

, (6.14)

with wb being the beam waist parameter. Furthermore, a constant orientation of ~u is assumed across the whole light spot, which is experimentally not given in general. But since no other information about the distribution of~u is available, this assumption is the most reasonable one. The final transmitted power P_t results from an integration across the illumination profile at the sample surface, weighted with the local transmittance T:

P_t= Z

T(n)·I dA . (6.15)

For comparison with experimental data, we normalize the electro-optically induced change ofPtto the power at zero fieldPt|_E=0and to the potential differenceU = Φtip−Φ_bsbetween the tip and the back electrode, as already defined for the experiments by Eq. 4.5.

6.3.2 Application to domain imaging

All modelling results presented in the following were derived by using the bulk values of the dielectric constants εa, εc, refractive indices no,ne, and electro-optic constants r₁₁₃^{ef f}, r₁₃₁^{ef f}, r^{ef f}₃₃₃ (unclamped, low frequency), listed in Tab. 3.1. At the sample surface, these values are typically modified as predicted by theory [84]. But as long as no complete set of experimentally validated parameters is available, we prefer using bulk values, even though this may give rise to additional errors of the numbers obtained.

The beam waist parameter was set to a fixed value wb = 1µm, while the tip radius Rtip, meniscus radius Rmen, tip-sample distance d, tilt angle φ of ~u within the xz plane, and lateral shiftss_x,s_y of the illumination profile with respect to the tip apex were varied.

We first discuss the influence of the water adsorption layer and meniscus on the electro-optic-response signal in order to find a range of realistic values of these parameters. Section 6.2.4 already showed that they have strong impact on the electric-field distribution. Since this is the electric field that probes the local electro-optic properties, there is a strong influence on the electro-optic response, too. Again, we consider a c domain, allowing the electric-field calculation to be performed in reduced 2D geometry. However, a sharp 180° domain wall breaks the cylindrical symmetry as far as the electro-optic response is concerned, and therefore the calculation of the electro-optic signal near such a domain boundary requires a complete 3D treatment.

The signal as calculated across a 180°c domain wall is shown in Fig. 6.6 for two different radii R = R_tip = R_men = R_contact of R =100 nm and R =150 nm, and for all three configurations of ambient media (meniscus, water only, air only). The other parameters were set to d = 5 nm, φ = 0, and s_x = s_y = 0. For both values of R, Fig. 6.6 reveals a significantly enhanced resolution for the meniscus case as compared to the air-only and water-only cases. This is a direct consequence of the additional electric-field confinement caused by the meniscus as described in Sec. 6.2.4. The experimentally determined resolu-tion shown in Fig. 4.7 is well reproduced by the model only if meniscus formaresolu-tion is taken into account. Also the absolute numbers indicate that water must be present between the tip and the sample. The electro-optic signal calculated with only air as ambient medium is much lower than the experimentally observed one, while the values for the water-only and meniscus case fit quite well to the typical experimental value of 5·10⁻⁶/V. Therefore, all subsequent calculations were performed under the assumption that a water layer is present and a meniscus is formed with R= 100. . .150 nm.

We now extend the modelling to a complete domain structure with mixedaandcdomain areas. The assumed domain geometry is illustrated in Fig. 6.7. A stripelikeadomain with a width of 2 µm is embedded into a structure of antiparallel c domains. This geometry now makes it necessary to perform the calculation of the electric-field distribution in a complete 3-dimensional treatment.

6.3. Electro-optic response modelling 65

electro-optic response [10-6 / V]

domain wall position [nm]

electro-optic response [10-6 / V]

domain wall position [nm]

Fig. 6.6 Effect of water layer and meniscus formation on the electro-optic re-sponse as modelled for a sharp 180°c domain wall. For both consid-ered values of the characteristic radius R=R_tip =R_men=R_contact, the resolution is enhanced significantly by the meniscus. With regard to both resolution and absolute numbers of the electro-optic response, good agreement with the experiment is only achieved if a water layer forming a meniscus is included in the model.

x z y

c domain embedded

a domain

Fig. 6.7 Mixed domain structure assumed for modelling. A stripelike a domain with a width of 2µm is embedded into a structure of antiparallel c do-mains. The walls between the c and a domains run at an angle of 45° into the sample. In accordance with the orientation of the adjacent c domain, the middle stripe contains two antiparallel a domains.

Because of the increased requirements of a 3D calculation for computational resources, we can afford only a limited number of variations of the geometric parameters. The electric-field modelling is therefore done for two parameter sets only, assuming againR= Rtip =Rmen =Rcontact with R = 100 nm and R = 150 nm, respectively, at a tip-sample separation d= 5 nm. Once we have calculated the electric-field distribution, we can use it to determine the electro-optic response for different sets of optical parameters.

First, we model the electro-optic response with a simple set of optical parameters given by: beam waist wb = 1 µm, tilt angle φ = 5°, and zero lateral shifts sx = sy = 0. The resulting electro-optic response signal across the domain structure is shown in Fig. 6.8 for

both values of R. Compared to the experimental data reported in Sec. 4.3.1, the model underestimates the a domain signal, while the c domain signal is reproduced quite well.

As seen from the color scale bar in Fig. 6.8, the electro-optic-response signal is stronger in case of R= 150 nm than for R= 100 nm. The lateral resolution, as deduced from the line profiles in the bottom part of Fig. 6.8, is found to be in the order of 2R for both cases, so it is higher for R = 100 nm. It is remarkable that the resolution is equal for a and cdomain transitions. This is generally not found in the experiment.

Figure 6.8 shows also a very pronounced electro-optic response decorating thea-cdomain boundaries. This can be attributed to an extraordinary electric-field distribution close to the boundary, resulting from the changed anisotropy orientation of the dielectric tensorε.

It manifests itself as two rims of opposite sign at the boundaries between the a domain and the two adjacentcdomains. Similar effects can be found in experimental data (Figs.

4.6, 4.8, 4.13), where however the decoration being in phase with theadomain response is typically not visible because it may be similar in magnitude to the dominating adomain signal. This is not perfectly reflected in the model, probably due to underestimation of theadomain response.

domain wall position [nm] domain wall position [nm]

-6electro-optic resp. [10/V]electro-optic response electro-optic response-6electro-optic resp. [10/V]

modelled, R = 150 nm modelled, R = 100 nm

Fig. 6.8 Modelled electro-optic response at a mixed domain structure. The a domain signal is underestimated, while the c domain signal is re-flected correctly, as suggested by comparison with experimental data.

Compared with the results for R=100 nm, the electro-optic response is increased and lateral resolution is decreased for R=150 nm. The pronounced electro-optic response at the a-c domain boundaries indi-cates an extraordinary electric-field distribution at these locations.

6.3. Electro-optic response modelling 67

To isolate the contribution of a specific electric-field component, we repeat the calculation of the electro-optic signal with all other components set to zero. Furthermore, we set also the electro-optic coefficients of either the aor thecdomains to zero, so that the response of only one domain type becomes visible. By this, we can elucidate how the two domains at ana-cdomain boundary contribute to the signal. As there are two relevant electric-field components E_x and E_z for the selected optical polarization orientation (Tab. 6.1), and two domain orientations aand c, the calculation splits up into four cases.

Figure 6.9 shows the modelling results obtained for R = 150 nm and φ = 5^◦. For easy comparison, all images are plotted with the same color scale, already used in Fig. 6.8.

The center image of Fig. 6.9 includes all contributions, while the outer images correspond to the four limiting cases as indicated. For the present set of parameters we find that for areas away from a-c domain boundaries the electro-optic response for both domain orientations stems only from the E_z component of the electric field. In contrast to this, very pronounced contributions stemming from the lateral field component E_x are found close to the a-c boundary. They are produced by the adomain as well as the c domain, with different magnitudes. They can be attributed to the special electric-field distribution at the boundary, which is characterized by an average lateral field component of significant magnitude across the optically addressed area.

E only _x (a domain)

E only z

(a domain)

E only _x (c domain)

E only z

(c domain)

Fig. 6.9 Contributions of the different domain types and electric-field compo-nents to the electro-optic response. The normal component of the electric field (Ez) determines the electro-optic response inside the do-mains, while lateral field components (E_x) give rise to significant con-tributions close to the a-cdomain boundaries. This decoration of the domain walls stems from the electro-optic response of both the a and the c domains.

The pronounced electro-optic response at the domain boundaries demonstrates that lateral components of the electric field can effectively contribute to the signal, provided that their average across the optically addressed volume does not vanish. This gets generally true for all domain areas – not only the boundaries – if the optical spot is shifted against the tip apex. Then, the electro-optic changes induced by the lateral components of electric-field distribution are weighted in an asymmetric way by the illumination profile. If these changes are probed by the chosen state of optical polarization, they will have an influence on the electro-optic response signal.

To show how a lateral shift of the spot modifies the signal for various orientations of the optical polarization (given by ~u(φ)), we arrange the results in a “matrix” shown in Fig.

6.10, with columns corresponding to different spot shiftss_x = (−500,0,500) nm and rows corresponding to different anglesφ= (−5,0,5)^◦. The underlying electric-field distribution was again calculated for a characteristic radius R = 150 nm and a tip-sample distance d= 5 nm.

The dependence on ~u(φ) is a consequence of the probing of different elements of the η matrix as indicated in Tab. 6.1. For normal incidence (φ= 0^◦), onlyη_xx is probed, and for each domain type only one field component and the corresponding electro-optic coefficient play a role. If φ6= 0 is selected, thenη_zz,η_xz, and η_zx are probed as well. Contributions stemming from diagonal elements (η_xx,η_zz) are affected only by the absolute value of φ, while contributions arising from off-diagonal elements (η_xz,η_zx) show an additional depen-dence on the sign of φ. All contributions to the electro-optic response can be associated with certain elements of the η matrix and the related electro-optic coefficients and field components.

Forφ= 0^◦, the electro-optic response of thecdomain area stems only fromη_xx =r₁₁₃E_z. Its value decreases if a lateral spot shift s_x is introduced because of misalignment of the electrically and optically addressed areas. Since the electro-optic response of theadomain area stems from ηxx =r333Ex, no signal is obtained in general for zero spot shift sx = 0 because Ex has an average value of zero across the light spot. Only close to the domain boundaries is a significant value found because of the special electric-field distribution present here. If a spot shift sx6= 0 is introduced, either positive or negative values of Ex

get favored by the “optical” weighting by the illumination, and therefore an electro-optic response arises across the whole a domain area. If s_x is replaced by −s_x, then also the favored sign of E_x is reversed and, hence, the electro-optic response changes its sign.

In case ofφ6= 0, additional elements ofηwith a different dependence onφare probed. For thecdomain area, a small contribution ofηzz =r333Ez is expected, which is independent of the sign of φ. A much stronger contribution is provided by η_xz =η_zx=r₁₃₁E_x, which is connected with a lateral field component again. At zero spot shifts_x = 0, a noticeable contribution is therefore found only close to the domain boundary, as already discussed above for the case of an a domain. If a spot shift s_x 6= 0 is introduced, the electro-optic response will be altered across the entire c domain area. In accordance with the probed element η_xz =η_zx =r₁₃₁E_x, this contribution depends on the signs of φand s_x.

6.3. Electro-optic response modelling 69

x x

x x

z y y

y y z

z z

s = -500 nmX s = 0 nmX s = +500 nmX

f = 5°

f = -5°

f = 0°

spot shift vs.

tip apex (X)

tilt f in xz-plane

frame size: 6 µm x 6 µm -2 [10 /V^-5 ] +2

electro-optic response scale:

domain geometry:

Fig. 6.10 Effect of spot shift and tilt on electro-optic response imaging. Depend-ing on the angle φ, different elements of the η matrix are probed. A spot shift s_x with respect to the tip apex leads to an effective contri-bution of elements connected with lateral field components.

Therefore, the electro-optic response of acdomain can be enhanced or suppressed or even overcompensated compared to the case φ = 0^◦. In the a domain area, ηzz =r113Ex and ηxz = ηzx = −r₁₃₁Ez become probed additionally in case of φ 6= 0. The first element depends on the lateral field component and behaves similar to theηxx element in the case φ= 0, while the second element depends also on the sign of φand is connected with the normal field componentE_z. It delivers the main contribution to the electro-optic response of theadomain area and decreases slightly with increasing absolute value of the spot shift, similar to the η_xx component in the cdomain case.

In summary, different tilt angles φand lateral spot shifts sx give rise to various combina-tions of signal contribucombina-tions, providing a variety of possible contrasts between the different domain orientations.

6.3.3 Application to ferroelectric hysteresis

The electric-field distribution produced by the scanning probe tip has a twofold function in case of the electro-optic inspection of ferroelectric switching characteristics. First, a sufficiently high electric-field component along the polar axis of the ferroelectric sample is required for domain switching. Second, the electric field strength is modulated by a much smaller value to probe the state of polarization averaged across the relevant sample volume. Both functions are provided by the same distribution of the electric field.

Switching occurs between the two antiparallel c domain orientations, that means fer-roelectric polarization is always aligned normal to the sample surface in the switching experiments. Therefore, the normal component E⊥ is the most important one for do-main switching. For the c domain geometry, this is also the relevant field component for electro-optic polarization detection, at least as long as the tilting angleφand lateral shifts s_x, s_y of the illumination are small. Since the experiments are performed at different

Switching occurs between the two antiparallel c domain orientations, that means fer-roelectric polarization is always aligned normal to the sample surface in the switching experiments. Therefore, the normal component E⊥ is the most important one for do-main switching. For the c domain geometry, this is also the relevant field component for electro-optic polarization detection, at least as long as the tilting angleφand lateral shifts s_x, s_y of the illumination are small. Since the experiments are performed at different

Im Dokument Local-scale optical properties of single-crystal ferroelectrics (Seite 71-85)